Linear independence measure of logarithms on a commutative algebraic group (Q2778029)

Let \(G\) be a commutative algebraic group over a number field \(K\). Denote by \(t_{G}\) the \(K\)-vector space tangent at the origin of \(G\) and by \(\exp:t_{G}(\mathbb C)\rightarrow G(\mathbb C)\) the exponential map of the complex Lie group \(G(\mathbb C)\). Fix a norm on the complex vector space \(t_{G}(\mathbb C)\) and denote by \(d\) the associated distance. Let \(u\in t_{G}(\mathbb C)\) satisfy \(\exp u\in G(K)\). A corollary of the main theorem solves one of the outstanding open problems on Diophantine approximation on algebraic groups: there exists a positive constant \(c\) such that, for any hyperplane \(W\) of \(t_{G}\) not going through \(u\), we have NEWLINE\[NEWLINE \log d(u,W)\geq -c\max\{1,h(W)\}, NEWLINE\]NEWLINE where \(h(W)\) is the absolute logarithmic Schmidt height of \(W\). NEWLINENEWLINENEWLINESuch an estimate, which is optimal in terms of \(h(W)\), had been obtained only in special cases: firstly for linear groups by Baker-Feldman at the end of 1960's, secondly for a power of an elliptic curve with complex multiplication by \textit{M. Ably} in 2000 [Ann. Inst. Fourier 50, No. 1, 1-33 (2000; Zbl 0957.11030)] and thirdly for products of elliptic curves by \textit{S. David} and \textit{N. Hirata-Kohno} more recently [Wüstholz, Gisbert (ed.), A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press, 26--37 (2000; Zbl 1041.11053) and Linear forms in elliptic logarithms (manuscript)]. NEWLINENEWLINENEWLINEThe main theorem of the paper under review is sharper than the above-mentioned corollary since the dependence of \(c\) in several parameters (including \(u\) and the degree of the number field \(K\)) is explicitly given, improving earlier estimates due to \textit{N. Hirata-Kohno} [Invent. Math. 104, 401-433 (1991; Zbl 0716.11035)]. The main theorem also studies the case where \(W\) goes through \(u\), giving an upper bound for the degree of an algebraic subgroup \(\widetilde{G}\) of \(G\) such that \( t_{\widetilde{G}}\subset W\) and \(u\in t_{\widetilde{G}}(\mathbb C)\).